reserve X,Y,Z for set,
        x,y,z for object,
        A,B,C for Ordinal;
reserve U for Grothendieck;

theorem Th19:
  for X be infinite set holds not X in Tarski-Class {X}
proof
  let A be infinite set;
  deffunc Bool(set,set) = $2\/bool $2;
  consider f be Function such that
A1: dom f = NAT & f.0 = {{A},{}} and
A2: for n be Nat holds f.(n+1) = Bool(n,f.n) from NAT_1:sch 11;
  set U = Union f;
  f.(0+1) = f.0 \/ bool (f.0) & {} c= {{A},{}} by A2;
  then
A3: {} in f.1 by A1,XBOOLE_0:def 3;
  then
A4: {} in U by A1,CARD_5:2;
  defpred Min[object,object] means  $1 in f.$2 & $2 in dom f &
    for i,j be Nat st i < j =$2 holds not $1 in f.i;
A5: for x be object st x in U ex y be object st Min[x,y]
  proof
    let x such that
A6: x in U;
    consider k be object such that
A7: k in dom f & x in f.k by A6,CARD_5:2;
    reconsider k as Nat by A7,A1;
    defpred M[Nat] means x in f.$1;
    M[k] by A7;
    then
A8: ex k be Nat st M[k];
    consider k be Nat such that
A9: M[k] and
A10:  for n being Nat st M[n] holds k <= n from NAT_1:sch 5(A8);
    take k;
    thus thesis by A10,A9,A1,ORDINAL1:def 12;
  end;
  consider Min be Function such that
A11:dom Min = U & for x be object st x in U holds Min[x,Min.x]
    from CLASSES1:sch 1(A5);
A12:U is subset-closed
  proof
    let X,Y such that
A13:  X in U & Y c= X;
    set x=Min.X;
A14:  Min[X,x] by A11,A13;
    then reconsider x as Nat by A1;
    per cases by NAT_1:3;
    suppose X=Y;
      hence thesis by A13;
    end;
    suppose x=0 & X<>Y;
      then X ={A} or X={} by A1,TARSKI:def 2,A14;
      hence thesis by A4,A13,ZFMISC_1:33;
    end;
    suppose x >0;
      then reconsider x1=x-1 as Element of NAT by NAT_1:20;
      x1 < x1+1 by NAT_1:13;
      then
A15:    not X in f.x1 by A11,A13;
A16:    f.(x1+1) = f.x1 \/ bool (f.x1) by A2;
      then X in bool (f.x1) by A14,A15,XBOOLE_0:def 3;
      then X c= f.x1;
      then Y c= f.x1 by A13;
      then Y in f.x by A16,XBOOLE_0:def 3;
      hence thesis by A14,CARD_5:2;
    end;
  end;
A17: for X st X in U holds bool X in U
  proof
    let X such that
A18:  X in U;
    set x=Min.X;
A19:  Min[X,x] by A11,A18;
    reconsider x as Nat by A19,A1;
A20:  f.(x+1) = f.x \/ bool (f.x) by A2;
    per cases by NAT_1:3;
    suppose
A21:    x=0;
      then X ={A} or X= {} by A1,TARSKI:def 2,A19;
      then bool X c= {{A},{}} by ZFMISC_1:7,ZFMISC_1:24,ZFMISC_1:1;
      then bool X in f.1 by A20,A21,A1,XBOOLE_0:def 3;
      hence thesis by A1,CARD_5:2;
    end;
    suppose x >0;
      then reconsider x1=x-1 as Element of NAT by NAT_1:20;
      x1 < x1+1 by NAT_1:13;
      then
A22:  not X in f.x1 by A11,A18;
A23:  f.(x1+1) = f.x1 \/ bool (f.x1) by A2;
A24:  f.(x+1) = f.x \/ bool (f.x) by A2;
      X in bool (f.x1) by A23,A19,A22,XBOOLE_0:def 3;
      then bool X c= bool (f.x1) by ZFMISC_1:67;
      then bool X c= f.x by A23,XBOOLE_1:10;
      then bool X in f.(x+1) by A24,XBOOLE_0:def 3;
      hence thesis by A1,CARD_5:2;
    end;
  end;
  defpred D[Nat] means f.$1 is finite;
A25: D[0] by A1;
A26: for n be Nat st D[n] holds D[n+1]
  proof
    let n be Nat such that
A27:  D[n];
    f.(n+1) = f.n \/ bool (f.n) by A2;
    hence thesis by A27;
  end;
A28: for n be Nat holds D[n] from NAT_1:sch 2(A25,A26);
A29: for x be set st x in dom f holds f.x is countable
  proof
    let x be set such that
A30:x in dom f;
    reconsider x as Nat by A30,A1;
    D[x] by A28;
    hence thesis;
  end;
  then
A31:U is countable by CARD_4:11,A1;
  for X holds X c= U implies X,U are_equipotent or X in U
  proof
    let X such that
A32:  X c= U;
    per cases;
    suppose card X = omega or X={};
      then card U=card X or X={}
        by A32,CARD_1:11,CARD_3:def 14,A29,CARD_4:11,A1;
      hence thesis by A3,A1,CARD_5:2,CARD_1:5;
    end;
    suppose
A33:  card X <> omega & X<>{};
      then card X c< omega by A32,CARD_3:def 14,A31;
      then card X in omega by ORDINAL1:11;
      then
A34:  X is finite;
      defpred P[object,object] means $1 in f.$2 & $2 in dom f;
A35:  for x be object st x in X ex u be object st P[x,u]
      proof
        let x;assume x in X;
        then ex n be object st n in dom f & x in f.n by A32,CARD_5:2;
        hence thesis;
      end;
      consider fX be Function such that
A36:  dom fX = X and
A37:  for x be object st x in X holds P[x,fX.x] from CLASSES1:sch 1(A35);
A38:  now let y be object;
        assume y in rng fX;
        then ex x be object st x in dom fX & fX.x=y by FUNCT_1:def 3;
        hence y is natural by A1,A37,A36;
      end;
      reconsider RNG=rng fX as finite natural-membered non empty set
        by A33,RELAT_1:42,A38,MEMBERED:def 6,A36,A34,FINSET_1:8;
      set m= max RNG;
      m in omega by ORDINAL1:def 12;then
      reconsider m as Nat;
A39:  f.(m+1) = f.m \/ bool (f.m) & m+1 in omega by A2;
      X c= f.m
      proof
        let x be object such that
A40:    x in X;
        P[x,fX.x] by A40,A37;
        then reconsider k=fX.x as Nat by A1;
        k in rng fX by A40,A36,FUNCT_1:def 3;
        then
A41:    k <= m by XXREAL_2:def 8;
        defpred Q[Nat] means x in f.$1;
A42:    Q[k] by A40,A37;
A43:      for i be Nat st k <= i & Q[i] holds Q[i+1]
        proof
          let i be Nat such that
A44:      k <= i & Q[i];
          f.(i+1) = f.i \/ bool (f.i) by A2;
          hence thesis by A44,XBOOLE_0:def 3;
        end;
        for i be Nat st k <= i holds Q[i] from NAT_1:sch 8(A42,A43);
        hence thesis by A41;
      end;
      then X in f.(m+1) by A39,XBOOLE_0:def 3;
      hence thesis by A1,CARD_5:2;
    end;
  end;
  then
A45: U is Tarski by A12,A17;
  {A} in f.0 & 0 in omega by A1,TARSKI:def 2;
  then U is_Tarski-Class_of {A} by A45,CARD_5:2,A1;
  then
A46: Tarski-Class {A} c= U by CLASSES1:def 4;
  not A in U
  proof
    assume
A48:  A in U;
    then
A49:  Min[A,Min.A] by A11;
    then reconsider x=Min.A as Nat by A1;
    per cases by NAT_1:3;
    suppose x=0;
      hence thesis by A1,A49;
    end;
    suppose x >0;
      then reconsider x1=x-1 as Element of NAT by NAT_1:20;
      x1 < x1+1 by NAT_1:13;
      then
A50:  not A in f.x1 by A48,A11;
A51:  f.(x1+1) = f.x1 \/ bool (f.x1) by A2;
      A in bool (f.x1) by A51,A49,A50,XBOOLE_0:def 3;
      then A c= f.x1 & f.x1 is finite by A28;
      hence thesis;
    end;
  end;
  hence thesis by A46;
end;
